New analytic techniques in differential and difference systems

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

New methods based on generalized Borel summation and trans-asymptotic expansions are being developed by a number of analysts and are emerging as a new field of study, "analyzability theory". The methods enable one to treat solutions of equations at singularities much more general then usually dealt with and provide a rigorous foundation to previously formal calculations. The present proposal aims at developing and applying these new techniques. The PIs will continue their research in the context of ODEs, difference equations and PDEs. Applications addressed include the classification of nonlinear ODEs at irregular singularities, determining their integrability properties as well as their reducibility to simpler systems, study of adiabatic invariance, and Painleve' analysis for difference equations. These techniques will be used to continue rigorous research of the time dependent Schroedinger problem under periodic perturbations of arbitrary strength and also of resonance theory. The project is a piece in an unfolding enterprise to bridge the gap from formal calculations to rigorous proofs, and in the advance of what appears to be emerging as a general theory of singular behavior. Countless applications exist in mathematics as well as in the study of physical phenomena.